I think there are two key misconception you seem to be having here. Before I say what it is, a disclaimer: you haven't filled in your profile, so I have no idea of your level of knowledge. Therefore, if my answer seems to be telling you how to suck eggs, please be aware that I am not meaning to be condescending.
1) You seem to be thinking of the "Big Bang" as an explosion that scatters everything in the Universe far and wide to get to their positions today. This is not accurate.
2) Stars have been outputting light constantly since the big bang. So even in your "explosion" model, there would be no contradiction. You seem to be imagining all the light in the universe being created at once and then "outrunning" everything aways from the explosion centre. But imagine two observers coasting in diverging but roughly parallel paths: observer $A$ to shine a light *at any time after the "explosion" that reaches $B$ sometime afterwards. $B$ does not need to wonder "how did I get here before the light did?". $B$ simply calculates where $A$ was when they sent the light pulse and knows it has to be after the explosion.
Point (2) is still a problem for the cosmic microwave background radiation - it doesn't explain how we can still see the CMBR, which indeed began its existence with the Big Band. But then we return to point A. The "big bang" "created" space and time and the spacetime manifold "expanded" afterwards: extremely swiftly at first, then it slowed down and now we believe it may be speeding up again. I'm always wary of analogies, but the following is a good possible lower dimensional analogy for the geometry. Imagine the universe as a balloon - a 2-sphere. But you need to imagine that there is nothing aside from the balloon. At some point, the balloon is shrunken to a very small point, and then swells. Points on the 2-dimensional universe are simply borne along on its surface. As the universe expands, light beginning at different points reaches a particular point at all times: it's simply that the points of origin of possible points whose light reaches us today changes as time goes on. Ponder this hackneyed phrase: *if we ask where the position of the big bang is today, the answer is "everywhere", for all points were united, or at least exquisitely near to one another, at the time of the Big Bang".